Dead Reckoning

Dead reckoning is the most fundamental navigation technique: estimate where you are based on where you were, how fast you've been going, and which direction. It precedes celestial navigation, precedes Harrison's chronometer, precedes GPS by millennia. Every other navigation method ultimately fits within a DR framework — the fix methods correct the DR estimate.

This article covers the math, the historical instruments, the error model, and the modern equivalents. Companion to /learn/celestial-navigation, which describes the fix method that historically supplemented dead reckoning at sea.

The basic math

Starting from a known position (a “fix”), the navigator records:

• The course (true compass heading) being steered.
• The speed being made through the water.
• The time elapsed.

The dead-reckoning position is computed by applying the course-and-speed vector to the previous position:

new_latitude  = old_latitude  + (speed × cos(course) × time) / R_lat
new_longitude = old_longitude + (speed × sin(course) × time) / (R_lon × cos(latitude))

Where R_lat = 60 nautical miles per degree of latitude and R_lon is similarly 60 nautical miles per degree of longitude at the equator (with the cos(latitude) correction at higher latitudes — see /learn/what-is-longitude).

Example: starting from 30°00' N 050°00' W, course 090° true (due east), speed 10 knots, elapsed 6 hours.

• Distance run: 10 × 6 = 60 nautical miles.
• Change in latitude: cos(90°) × 60 = 0 nm (no N-S change).
• Change in longitude: sin(90°) × 60 / cos(30°) = 60 / 0.866 = 69.3' of longitude eastward.
• New position: 30°00' N 048°50.7' W.

That's the basic DR computation. Real navigators use charts and plotters rather than equations — the math is built into the chart projections and plotting instruments.

Drift, leeway, and current

The simple DR equation assumes the ship moves exactly along the steered course at exactly the measured speed. In reality:

• Current (ocean current, tidal current, river current) pushes the ship sideways or along the heading.
• Leeway is the angle between the heading and the actual course made good — caused by wind pushing the ship.
• Sea state introduces both random and systematic errors.
• Steering error — the ship doesn't exactly hold its course; helmsman variations and autopilot deviations add up.

The full DR equation accounts for these:

course_made_good = steered_course + leeway_angle
position_after_DR  = old_position + (speed × course_made_good × time)
position_after_set = position_after_DR + (current_set × current_drift × time)

This converts the DR estimate to an estimated position (EP) — a more refined guess than the raw DR. The EP is plotted on the chart as an open circle with cross-hairs; the DR is plotted as a half-circle. Distinct symbols, distinct meanings.

Historical instruments

Dead reckoning predates writing. Ancient Phoenicians, Polynesians, Norse, Chinese, and Arab seafarers all practised forms of DR. The European-tradition instruments of the 14th–19th centuries are the most documented.

The magnetic compass

Introduced to European navigation in the late 12th century (earlier in China). A magnetized needle on a low-friction pivot points to magnetic north. The compass is divided into 360° (modern) or 32 named points (historical — north, north-by-east, north-northeast, etc.).

The compass replaced star-sighting (Polaris in the north) as the primary heading reference, especially under cloudy skies. Magnetic compass readings must be corrected for:

• Variation — the local angle between true north and magnetic north, varying by location and time (see related material at the IGRF / WMM models).
• Deviation — local error from the ship's magnetic field (iron in the hull, electrical wiring). Each compass on each ship has a unique deviation table.

The chip log

A wooden chip (panel of wood, weighted to float vertically) attached to a long line. The line is marked with knots at regular intervals. The procedure: throw the chip overboard, let the line run free for a measured interval (typically 28 seconds, timed by a sandglass), count the knots that passed overboard.

The knot spacing is chosen so each knot represents one nautical mile per hour:

1 nm/h × (28 s / 3600 s/h) = 28/3600 nm ≈ 47.25 feet

So the knots are spaced 47.25 feet apart (modern equivalent; historical values varied slightly with sandglass calibration). A reading of “10 knots passed” meant 10 nm/h.

This is the origin of the unit “knot” for ship speed. The chip log was used continuously from the 16th century until the patent log (a propeller-driven mechanical log) displaced it in the 19th century.

The traverse board

A round wooden board, ~30 cm diameter, with eight pegs around the compass rose marked at half-hour intervals (one “watch” was eight half-hours = 4 hours). The peg position recorded the course steered during each half-hour; a separate column of pegs recorded the speed (in knots).

At the end of each watch, the helmsman's record was transferred to the ship's log book. The traverse board was a simple solution to the problem of recording multiple data points over a long watch with limited writing materials.

The sandglass

A hourglass-type sand timer, used to measure the chip-log interval (28 seconds, typically) and to mark half-hour and hour intervals for watch-changing. Ship's bells (one to eight bells per watch) corresponded to sandglass turns.

The error model

Errors in dead reckoning accumulate over time. The error budget for a 24-hour DR run, typical values:

| Source | Per-hour error | 24-hour cumulative | | ----------------------- | --------------- | ------------------ | | Compass heading | ±2° | ±2° | | Helmsman steering | ±2° | ±2° | | Speed measurement | ±0.5 knot | ±12 nm | | Current (uncorrected) | ±0.5 knot | ±12 nm | | Leeway | ±1° | ±1° | | Total cross-track | | ±10–15 nm | | Total along-track | | ±10–15 nm |

A skilled navigator running a 24-hour DR track expects to be within 5–15 nautical miles of the true position. After 72 hours without a fix, the error envelope grows to ~30–50 nm.

Errors grow linearly with time in the simple model, but in practice errors are correlated (a 1° heading bias persists across watches), so the actual error grows more like t^1.2 to t^1.5 — slightly faster than linear.

Dead reckoning is therefore self-limiting: at some interval, the navigator must take a fix and reset the DR plot. Historical practice: noon-sun-sight for latitude every day (the daily “noon fix”), supplemented by star sights at twilight. Modern practice: GPS fix every few seconds.

A 24-hour DR example

Consider a steam ship leaving New York at 1200 UTC on a known date, bound for Liverpool, course 090° (initial), speed 15 knots:

• 1200 (start): fix from harbour pilot, 40°42'N 074°00'W.
• 1800: 6 hours × 15 knots = 90 nm east → roughly 40°42'N 072°10'W (DR position).
• 0000 (next day): 12 hours × 15 knots = 180 nm east → roughly 40°42'N 070°23'W. Apply correction for current (Gulf Stream sets ~040° at ~2 knots): EP shifts ~12 nm to ENE.
• 0600: 18 hours × 15 knots = 270 nm east → roughly 40°42'N 068°33'W. Course adjusted (great-circle routing — see /learn/great-circle-distance) to 085° to compensate.
• 1200 (next day): 24 hours × 15 knots = 360 nm east → ~40°50'N 066°45'W. Take noon sun sight; sextant gives latitude 40°45'N — DR was 5' high.

The discrepancy at noon (5 nm in 24 hours) is well within the ±15 nm tolerance. The DR has held up. The fix updates the plot; DR resumes from the new fix.

Polynesian dead reckoning

The Polynesian non-instrument navigation tradition is arguably the most sophisticated pre-modern dead-reckoning method. Master navigators used:

• Star paths — memorized risings and settings of specific stars to maintain heading throughout the night.
• Wave patterns — the swell direction (set by distant winds, persistent for days) maintains heading by feel.
• Wind patterns — the trade winds' consistency.
• Birds and clouds — for proximity to land and reading weather.
• Mental dead-reckoning — the navigator maintains a continuous mental track of where the canoe is.

Polynesian voyages of 2,000+ nautical miles (Marquesas to Hawaiian Islands, ~3,500 nm) were navigated this way with arrival errors of perhaps 20–50 nm — comparable to European instrument-based dead reckoning of the same era. The Polynesian Voyaging Society maintains this tradition; modern voyages of Hōkūleʻa (replica double-hulled canoe) have circumnavigated the globe using only traditional methods.

Modern inertial navigation

The modern electronic equivalent of dead reckoning is inertial navigation. An inertial navigation system (INS) uses:

• Gyroscopes — measure rotation rate. Modern fibre-optic gyros or ring-laser gyros measure rotation to microradian-per-second precision.
• Accelerometers — measure linear acceleration in each axis.
• Integration — the system integrates rotation (heading) and acceleration (velocity) to predict position.

A modern aircraft INS drifts ~1 nautical mile per hour — a 1,000× improvement over paper dead reckoning. Submarine INS (deployed for months without GPS) drift even less. The U.S. Navy's ballistic-missile submarines use INS as the primary navigation system; GPS is used only when surfacing.

INS shares the same mathematical structure as paper DR: the position estimate is updated by integrating the velocity over time. The improvement is in the precision of the input measurements and the speed of integration (continuous, vs. discrete watch intervals).

GPS+DR hybrid systems

Modern bridge electronics integrate GPS and dead reckoning. Common architecture:

1. GPS provides absolute position fixes every 1–10 seconds.
2. INS or simpler DR (compass + speed) integrates between fixes.
3. A Kalman filter fuses the two, weighting each by its uncertainty.

When GPS is briefly unavailable (urban canyon, satellite geometry, antenna obstruction), the INS carries the position estimate for the gap. When GPS returns, the filter resyncs.

This hybrid approach is what every modern car's navigation system, every aircraft's flight management system, and every ship's bridge electronics do. The purely-GPS navigation that smartphone apps approximate is the exception; production systems all do GPS+DR.

Dead reckoning as GPS backup

Bridge officers on commercial and naval ships run paper DR plots in parallel with GPS. The explicit purpose:

• Detect GPS failure: if the GPS position diverges from the DR position by more than the expected DR error envelope, GPS is likely malfunctioning (or being spoofed — see /learn/gps-jamming-and-spoofing).
• Maintain navigation continuity: if GPS fails for hours, the DR plot is the only navigation track.

Standard practice on Royal Navy and U.S. Navy ships: DR plot maintained continuously, GPS-vs-DR comparison logged at every watch change. Many commercial fleets follow the same practice.

Common misconceptions

“Dead reckoning is obsolete.” It's not. Inertial navigation systems do electronic DR; bridge officers do paper DR as GPS backup; aircraft autopilots use DR between waypoints. Every position-tracking system includes a DR component.

“Dead reckoning is just an estimate.” It's a structured estimate with a known error model. The accumulated error after 24 hours is predictable (±10–15 nm). It's “just an estimate” in the same sense GPS is “just a position” — both have stated accuracy bands that the navigator must respect.

“DR errors are random.” Most DR errors are systematic — compass bias persists, speed-measurement bias persists, current sets continuously. Random errors roughly cancel out; systematic errors accumulate. Good DR practice catches systematic biases by comparing DR and fix positions over time.

“The chip-log knot is the same as the unit 'knot.'” Yes — the unit name comes from the instrument. One knot = one nautical mile per hour, and the chip-log's knotted line was the original speed measurement.

“Polynesian navigators didn't do dead reckoning.” They did — using mental tracking rather than paper plots, but mathematically the same process. Star paths gave heading; wave patterns gave heading-continuity; mental tracking integrated heading and estimated speed over time to predict position. The underlying logic is identical to chart-based DR.

“Inertial navigation is fundamentally different from dead reckoning.” It's not. INS measures rotation and acceleration; integration gives heading and velocity; further integration gives position. Same mathematical structure as paper DR. The improvement is input precision and integration speed.

“GPS replaced dead reckoning.” GPS supplements dead reckoning. Modern systems use GPS as the fix method and DR as the prediction-between-fixes method; both run continuously. When GPS fails (jamming, urban canyon, polar latitude with poor satellite geometry), DR carries the position estimate forward.